Thermal Corrections to R\'enyi Entropy in BMS Field Theory

In the study of three-dimensional flat holography, the BMS field theory manifests the infinite-dimensional BMS$_3$ symmetry, a powerful tool in elucidating numerous universal phenomena. This paper explores a certain low-temperature limit of the BMS field theory. The primary focus lies in the calculation of the thermal correction to the R\'enyi entropy of the single interval on the cylinder from the replica trick and the uniformizing map. As a double check, an alternative method calculating the entanglement entropy is introduced, with the entanglement first law and the modular Hamiltonian.